Composite G-codes over formal power series rings and finite chain rings
Composite G-codes over formal power series rings and finite chain rings
In this paper, we extend the work done on $G$-codes over formal power series rings and finite chain rings $\mathbb{F}_q[t]/(t^i)$, to composite $G$-codes over the same alphabets. We define composite $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G.$ We show that the dual of a composite $G$-code is again a composite $G$-code in this setting. We extend the known results on projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings to composite $G$-codes. Additionally, we extend some known results on $\gamma$-adic $G$-codes over $R_\infty$ to composite $G$-codes and study these codes over principal ideal rings.
Keywords:
Composite $G$-codes, Group rings, Finite chain rings, Formal power series rings $p$-adic integers,
___
- [1] R. L. Bouzara, K. Guenda, E. Martinez-Moro, Lifted codes and lattices from codes over finite chain rings, arXiv:2007.05871.
- [2] S. T. Dougherty, Algebraic coding theory over finite commutative rings, SpringerBriefs in Mathematics Springer (2017).
- [3] S. T. Dougherty, J. Gildea, A. Korban, Extending an established isomorphism between group rings and a subring of the n n matrices, International Journal of Algebra and Computation, Published: 25 February 2021.
- [4] S. T. Dougherty, J. Gildea, A. Korban, G- codes over formal power series rings and finite chain rings, J. Algebra Comb. Discrete Appl. 7 (2020) 55–71.
- [5] S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68, Advances in Mathematics of Communications 14(4) (2020) 677–702.
- [6] S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, Composite matrices from group rings, composite G-codes and constructions of self-dual codes, arXiv:2002.11614.
- [7] S. T. Dougherty, J. Gildea, R. Taylor, A. Tylshchak, Group rings, G-codes and constructions of self-dual and formally self-dual codes, Des. Codes, Cryptogr. 86(9) (2017) 2115–2138.
- [8] S. T. Dougherty, H. Liu, Y. H. Park, Lifted codes over finite chain rings, Mathematical Journal of Okayama University 53 (2011) 39–53.
- [9] S. T. Dougherty, H. Liu, Cyclic codes over formal power series rings, Acta Mathematica Scientia 31(1) (2011) 331–343.
- [10] J. Gildea, A. Kaya, R. Taylor, B. Yildiz, Constructions for self-dual codes induced from group rings, Finite Fields Appl. 51 (2018) 71–92.
- [11] T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math 31(3) (2006) 319–335.
- [12] B. R. McDonald, Finite rings with identity, New York: Marcel Dekker (1974).
- Başlangıç: 2015
- Yayıncı: İrfan ŞİAP
Sayıdaki Diğer Makaleler
The exact annihilating-ideal graph of a commutative ring
Subramanian VİSWESWARAN, Premkumar T. LALCHANDANİ
Composite G-codes over formal power series rings and finite chain rings
Algebraic methods in difference sets and bent functions
Pradipkumar H. KESKAR, Priyanka KUMARİ
On unit group of finite semisimple group algebras of non-metabelian groups of order 108
Gaurav MİTTAL, Rajendra. K. SHARMA
General degree distance of graphs